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User blog:Mh314159/new YIP notation
This is my most current version of YIP notation so far, make the best use I can of the excellent P進大好きbot's advice. Thank you to everyone who looks at this and has feedback. Rule set F a, b are nonnegative integers f0(a) = a + 1 fa(0) = fa-1(a) fa(b) = fa-1fa(b-1)(fa(b-1)) Example: f1(2) = f04(4) because f1(1) = f0f1(0)(f1(0)) = f0f0(1)(f0(1)) = f02(2) = 4, so f1(2) = 8 Example: f1(3) = f08(8) = 16 so f1(x) = (2)(x+1) Rule Set U Universal Relationships gp(x) = xgp-1(x)(x) g0(x) = xx(x) my(x) = my-1my(x-1)(x) my(0) = my-1(y) m0(x) = m-1m0(x-1)(x) m0(0) = m-1m(m) Rule set 1 One bracketed number a, n are nonnegative integers a = gnn(n) if a = 1, n = 1 if a > 1, n = a-1 00(x) = fx(x) Rule set 2 Many bracketed numbers drop trailing zeroes all entries contained in brackets are nonegative integers a,b,...,α,0 = a,b,...,α a,b,...,α,β with N terms = gnn(n) if a = 1, n = [b,...,α,β,b,...,α,β-1] ex: 0,1,2 = [1,2,1,1] if a > 1, n = a-1,b,...,α,β 00(x) = x,x,...,x,β-1 with N terms Proposed new Universal Relationships and Single Bracket Number: Using braces, I have allowed for an unlimited extension of functions similar to "g" above, in the form of {{m}n} which recurse to the underlined functions, now replaced by {{m}0} at which point the subscripts begin recursing and we eventually reach the base function {{0}0}0 which applies to each of the higher level rule sets. I think this makes the single bracket number faster than the string of numbers in the old notation. {{m}n}p(x) = {{m-1}n}{{m}n}p-1(x)(x) {{m}n}0(x) = {{m-1}n}{{m}n}0(x-1) (x) {{m}n}0(0) = {{m-1}n}m(m) {{0}n}y(x) = {{0}n}y-1{{0}n}y-1(x)(x) {{0}n}0(x) = {{x}n-1}x(x) {{m}0}y(x) = {{m}0}y-1{{m}0}y(x-1)(x) {{m}0}y(0) = {{m}0}y-1(y) {{m}0}0(x) = {{m-1}0}{{m}0}0(x-1)(x) {{m}0}0(0) = {{m-1}0}m(m) now, for other rule sets, use base function {{0}0}0(x) Rule set 1: Single bracketed number: 0 = 1 a = {{n}n}n(n) n = a-1 {{0}0}0(x) = fx(x) Here is a partial expansion of 1, until a number is reached that makes further numerical recursion not possible: \[ \{ \{ 1 \} 1 \}_1^1 (1) \] \[ \{ \{ 1 \} 1 \}_1^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{ \{ 1 \} 1 \}_0^{} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{1\}1\}_0 (0)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_1 (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{0\}1\}_0 (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_1 (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{1\}0\}_1 (0)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{1\}0\}_0 (1)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{0\}0\}_{\{\{1\}0\}_0 (0)} (1)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{0\}0\}_{\{\{0\}0\}_1 (1)} (1)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{0\}0\}_{\{\{0\}0\}_0^{\{\{0\}0\}_1 (0)} (1)} (1)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{0\}0\}_{\{\{0\}0\}_0^{\{\{0\}0\}_0 (1)} (1)} (1)} (1)} (1)} (1)}^{} (1) \] \[ \{ \{ 0 \} 1 \}_{\{\{0\}1\}_{\{\{0\}1\}_0^{\{\{1\}0\}_0^{\{\{0\}0\}_{f_2 (2)} (1)} (1)} (1)} (1)}^{} (1) \] \[ f_2 (2) > 2 \uparrow \uparrow \uparrow 4 \] And still a long way to go. There will be more than 2^^^4 levels of functional powers just on the expansion of {{0}0}f2(2)(1) alone, with the subscript on each level being f2(2)-2 Category:Blog posts